Air navigation services lost track of flight AH5017 about 50 minutes after take-off from Ouagadougou, the capital of Burkina Faso, at 3am UK time, the official Algerian news agency APS said.
It was en route to Algiers, in Algeria, when the crew asked Niger air control to adjust their route because of a storm in the area.
French President Francois Hollande cancelled a planned visit to overseas territories and said all military means on the ground would be used to locate the aircraft.
He said: "Everything must be done to find this plane. We cannot identify the causes of what happened."
Two French Mirage warplanes scoured the vast desert area around the northern Malian city of Gao for the aircraft, which had 51 French nationals on board.
French Foreign Minister Laurent Fabius said the Air Algerie flight had probably crashed.
Mali's President Ibrahim Boubacar Keita later said that wreckage of the flight had been spotted in his country's desert.
He said: "I have just been informed that the wreckage has been found between Aguelhoc and Kidal." He did not give any more details.
Air Algerie Flight 5017 was being operated by private Spanish airline Swiftair, the latter company said, adding that the plane was carrying 110 passengers and six crew, who are all Spanish nationals.
Swiftair said it had not been possible to make contact with the plane and was trying to ascertain what had happened.
The airline was quoted as saying: "In keeping with procedures, Air Algerie has launched its emergency plan."
The crew included two pilots and four cabin staff.
The list of passengers included 51 French people, 27 Burkina Faso nationals, eight Lebanese, six Algerians, five Canadians, four Germans, two Luxembourg nationals, one Swiss person, one Belgian, one Egyptian, one Ukrainian, one Nigerian, one Cameroonian and one Malian, Burkina Faso transport minister Jean Bertin Ouedraogo said.
The plane had been missing for hours before the news was made public.
It was not known why airline or government officials did not make it public earlier.
The flight path of the plane from Ouagadougou to Algiers was not immediately clear. Ouagadougou is in a nearly straight line south of Algiers, passing over Mali where unrest continues in the north.
Northern Mali fell under control of ethnic Tuareg separatists and then al Qaeda-linked Islamic extremists following a military coup in 2012.
A French-led intervention last year scattered the extremists, but the Tuaregs have pushed back against the authority of the Bamako-based government.
A senior French official said it seemed unlikely that fighters in Mali had the kind of weaponry that could be used to shoot down a plane.
The official said that they primarily have shoulder-fired weapons, which are not enough to hit a passenger plane flying at cruising altitude.
The MD-83 aircraft is part of the McDonnell Douglas MD-80 family of twin-engined jets that entered service in 1980.
A total of 265 of the MD-83 model were delivered before McDonnell Douglas, by then part of Boeing, halted production in 1999.
A spokesman for the manufacturer said: "Boeing is aware of the report.
"We are awaiting additional information."